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Branch-and-Cut-and-Price using Stable Set polytope inequalities for the Capacitated- Ring-Star Problem

Pierre Fouilhoux 1 Aurélien Questel
1 RO - Recherche Opérationnelle
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : Given a graph G = (V,E,A), V is the set of nodes which is partitioned into the set of depots N, the set of customers U and the set of Steiner nodes W. A ring-star is a bounded elementary cycle going through one depot v in N along with a subset of arc (i,j) in A where i is a client and j is a node of the cycle. The Capacitated Ring-Star Problem (CRSP) consists in finding a subset of ring-stars of minimal cost such that every edge belongs to a limited number of cycles. We propose a set partitionning formulation for this problem together with a column generation technique. We study the auxiliary problem and show that a classical algorithmic approach would not lead to an efficient pricing procedure. We propose to use a Branch-and-Cut algorithm based on a previous work to solve the auxiliary problem. We then strengthen our formulation with additional inequalities, including the well-known clique and odd-cycle inequalities. These inequalities are known to be very efficient, meanwhile, their dual costs are hard to be taken into account in the auxiliary problem. For that reason, there have been very few attempts to incorporate these inequalities in a Branch-and-Price algorithm. We present a method that allows to handle such inequalities using a Branch-and-Cut algorithm to solve the auxiliary problem. We also propose some reduction operations over the polytope associated with the master problem in order to speed up the separation algorithms. Finally, we present an efficient Branch-and-Cut-and-Price algorithm CRSP.
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Contributor : Lip6 Publications <>
Submitted on : Wednesday, October 14, 2015 - 2:59:24 PM
Last modification on : Friday, January 8, 2021 - 5:48:02 PM


  • HAL Id : hal-01215574, version 1


Pierre Fouilhoux, Aurélien Questel. Branch-and-Cut-and-Price using Stable Set polytope inequalities for the Capacitated- Ring-Star Problem. International Symposium on Combinatorial Optimization (ISCO 2014), Mar 2014, Lisbon, Portugal. ⟨hal-01215574⟩



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